Season each butterflied chicken breast with paprika, garlic, salt, and pepper on all sides. Lie each piece flat on a cutting board; place 2 slices of ham and 1 slice of cheese inside each. Fold over to seal the chicken breast. Insert a wooden toothpick at each end to seal.
Dredge chicken breasts in flour, egg, and then bread crumbs. Place on a baking sheet lined with parchment paper. Dot with vegetable oil and bake until chicken reaches 180 degrees F, about 10 minutes. Garnish with chopped parsley, if desired.